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Piya Tan has an essay on the topic here: Is Rebirth Immediate? (SD 2.17)
And a commentary and translation of the sutta here, Kutūhala,sālā Sutta, which includes "To Vaccha’s second and final question, the Buddha makes his famous statement on the intermediate state."
My paraphrase of the essay (if I've read it correctly) is that:
That essay totals about 15 pages -- you can read the details.
It quotes Ven. Sujato's commentary on The Kutūhala,sāla Sutta:
From this we can conclude that the Buddha, following ideas current in his time—for Vacchagotta was a non-Buddhist wanderer (paribbājaka)—accepted that there was some kind of interval between one life and the next. During this time, when one has “laid down” this body, but is not yet reborn in another, one is sustained by craving, like a flame tossed by the wind is sustained by air. The simile suggests, perhaps, that the interval is a short one; but the purpose of the simile is to illustrate the dependent nature of the period, not the length of time it takes. While a fire is burning normally, it is sustained by a complex of factors, such as fuel, oxygen, and heat. But when a tongue of flame is momentarily tossed away from the source fire, it can last only a short while, and in that time it is tenuously sustained by the continued supply of oxygen. Similarly in our lives, we are sustained by food, sense stimulus, and so on, but in the in-between, it is only the slender thread of craving that propels us forward. The difference is, of course, that the flame will easily go out, while the fuel of craving propels the unawakened inexorably into future rebirth.